SOME IDENTITIES INVOLVING THE EULER AND THE CENTRAL FACTORIAL NUMBERS

Weiipeng Zhang Department of Mathematics, The University of Georgia, Athens, GA 30602

(Submitted February 1996-Final Revision June 1996)

1. INTRODUCTION AND RESULTS

Let x be a real number with \x\<x/2. The Euler sequence E = (E2„), n = 1,2,..., is defined by the coefficients in the expansion of

„=o(2»)!" secx = Y^-x2".

That is, E0 = 1, E2 = 1, E4 = 5, E6 = 61, E% = 1385, El0 = 50521,.... These numbers arose in some combinatorial contexts, and were investigated by many authors. For example, see Lehmer [7] and Powell [8]. The main purpose of this paper is to study the calculating problem of the summation involving the Euler numbers, i.e.,

y E2aiE2a2 •••E2ak (],

a^t^-n (2^)1(2^)! ... (2%)! ' ^

where the summation is over all ^-dimension nonnegative integer coordinates (01?a2, ...,ak) such that ax +a2 + • • • +ak - n, and k is any odd number with k > 1.

This problem is interesting because it can help us to find some new recurrence properties for (E2rt). In this paper we use the differential equation of the generating function of the sequence (E2rj) to study the calculating problems of (1), and give an interesting identity for (1) for any fixed odd number k>l. That is, we shall prove the following main conclusion.

Theorem: Let n and m be nonnegative integers and k = 2m +1. Then we have the identity

I E2a,E2a, •••E2al[

a\ +a2 + ^ak __n(2al)\(2a2)\...(2ak)\ 1 m

= (k- l)\(2n)! ? H)/4'V(2/ft +1,2m - 2/ + l)E2n+2m_2i,

where t(nl k) are central factorial numbers. From the above theorem, we may immediately deduce the following.

Corollary 1: For any odd prime/?, we have the congruence

(0 (mod p), ifp = 1 (mod 4), [-2 (mod p), ifp = 3 (mod 4). V i -

Corollary 2: For any integer n > 0, we have the congruences (a) E2n+2 + E2„ = 0(mod6), (b) + 10£2„+2+9£2„ = 0(mod24), (c) E2n+6+E2„^0(mod42).

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2. PROOF OF THE THEOREM

In this section, we shall complete the proof of the theorem. First, we give an elementary lemma which is described as follows.

Lemma: Let F(x) = l / cosx . Then, for any odd number k = 2m + l>l, F(x) satisfies the dif-ferential equation

m (2m)lFk(x) = £c;.(w)F( 2 w-2 / )(x),

/=o

where F( r ) (x) denotes the r* derivative of F(x), and the constants c^m), i = 0,1,2, ...,m, are defined by the coefficients of the polynomial

m Gm(x) = (x + l2)(x + 32)(x + 52) - (x + (2m-1)2) = 5>/ ( w K~ 7 ' -

Note: The constants qQn) in the Lemma are special cases of the generalized Stirling numbers of the first kind, s^(n, k), introduced by Comtet [2], i.e.,

n (x - £0)(x - £ ) • • • (x - %n_x) = £ •%(", 0*'' •

7=0

Moreover, the constants cz-(/w) are, in fact, the central factorial numbers t(n, k) (see Riordan [9]). The inverse and similar numbers are treated in many important papers by Carlitz [3] and [4], and by Carlitz and Riordan [5]. For some generalizations, see Charalambides [6].

Now we prove the Lemma by induction. From the definition of F(x), and differentiating it, we may obtain

„# / x sinx _,f/ . cos3x-f 2sin2xcosx 2 1 Ff(x) = — — , F"(x) = 4 = —~ ,

cos x cos x cosrx cosx i.e.,

2F3(x) = F'Xx) + F(x) . (2)

This proves that the Lemma is true for m = 1. Assume, then, that it is true for a positive integer m=u. That is,

(2u)\F2u+l(x) = J^c^F^-^ix). (3) /=o

We shall prove it is also true for m = u +1. Differentiating (3), we have

(2u-hl)\F2u(x)F%x) = Xq(^ ( 2 w _ 2 / + 1 ) W ? ;=0

2u(2u +1)! F2u-\x)(FXx)f + (2i/ +1)! F2u(x)F"(x) = J q(w)F(2M-2,+2)(x). (4)

From the equality

4_l1(4x2 - l2)(4x2 - 32) • • • (4x2 - (2w -1)2) = £ t{2n +1,2k + l)x2^,

we get cjfc(w) = (-l)Ar4^(2w + l,2ii-2Jfc + l). (5)

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These numbers are tabulated in Riordan [9]. Using this expression and the recursive relation t(n, k) = t(n-2,k-2)-j(n- 2)2 t(n - 2, k), we have the recurrence relation

ck(n +1) = ck(n) + (2n +1) V i ( " ) , (6)

with initial conditions c0(n) = 1, cn(n) = 1232... (2n-1)2. Substituting (F '(x))2 by F 4 ( x ) - F 2 ( x ) and F"(x) by 2F3(x) - F(x) in (4) and applying (3) and (6), we have

(2u + 2)\F2u+3(x) = (2u)\(2u + l)2F2u+\x) + f^ci(u)F(2u+2~2i\x) ;=0

= (2i# + 1 ) 2 J ^.(w)F(2M-2/)(x) + J ^(w)^2M+2-2/>(x)

= c0(u)F(2u+2\x) + (2i# + \fcu{u)F{x) + £ (cM(u) + (2u + l ) 2 ^ ) ) ^ 2 " " 2 ' " ^ ) /=o

= c0(u + l)F(2*+2)(x) + ctt+1(w + l)F(x) + J c,(w + l)F(2w+2-2/)(x) u+l

= Zc/(w + 1)^( 2"+ 2"2 , ) (4

That is, the Lemma is also true for m - u +1 . This proves the Lemma. Now we complete the proof of the Theorem. Note that

F{2i)(x) = Y Eln+2i x2n i = 0 1 2

Comparing the coefficient of x2n on both sides of the Lemma and applying (5), we immediately obtain

Er, En Er> 1 m

1 m

= (2^)! ̂ ( _ 1 ) ' 4 ' r ( 2 ' M + l ' 2 m ~ 2 i + l^E^m-2i,

where the constants c^m), i - 0, 1,2,..., m are the coefficients of the polynomial m

GJx) = (x + l2)(x + 32)(x + 52) • • • (x + (2iff-1)2) = X C / W x ^ ' -7=0

This completes the proof of the Theorem.

Proof of the Corollaries; Taking n = 0 and k = p in the Theorem, and noting that EQ = 1, (p-1)1 = - 1 (modp) (Wilson's theorem, see Apostol [1]), we can get

- 1 . (p-1)! = £ ^ £ - 1 ^ ^ . ̂ + c^£zi^

= £p_1 + l2325272 . . . ( /? -2) 2 s ^ 1 + (-i)^(p_i)! .£^-(-1)^ (modpX

where we have used the congruences

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Therefore, _ JO (mod p)v ifp = 1 (mod 4),

p~l = [-2 (mod p\ ifp = 3 (mod 4).

This completes the proof of Corollary 1. Taking m-\ and 2 in the Theorem, respectively, we can get

^2n+A+E2n+2 = E2n+2 + E2n^0 (mod 2),

+ 1 0 £ 2 „ + 2 + 9 £ 2 ^ 0 (mod 24).

Thus, 0^E2n+4 + 10E2n+2+9E2n^E2„+4+E2„+2 = 0 (mod3). Since (2, 3) = 1, E2n+4 + E2n+2 = 0 (mod 2), we have E2n+4 + £2n+2 = 0 (mod 6), that is, E2n+2 +E2n = 0 (mod 6), n = 1,2, 3 , . . . .

Similarly, taking m - 4 in the Theorem, we can obtain the congruent equation

£2„+8 + 84JE2„+6 + 1974£2w+4 + 12916£2w+2 + 1 1 0 2 5 £ 2 ^ 0 (mod 40320).

Thus, 0 s £ ^ + 8 4 £ ^ + 1 9 7 4 ^ ^ + 1 2 9 1 6 ^ ^ + 1 1 0 2 5 ^ - £2„+8 + £2„+2 (mod21), that is, E2n+6+E2n = 0 (mod21), w = 1, 2, 3, . . . . Noting that E2n+6+E2n = 0 (mod 2) and (2,21) = 1, we get E2n+6+E2n = 0 (mod 42). This proves Corollary 2.

ACKNOWLEDGMENTS

The author expresses his gratitude to Professor Andrew Granville and to the anonymous referee for their very helpful and detailed comments.

REFERENCES

1. Tom M. Apostol. Introduction .to Analytic Number Theory. New York: Springer-Verlag, 1976.

2. L. Comtet. "Nombres de Stirling generaux et fonctions symmetriques." C. R. Acad. Sc. Paris, SerieA275(1972):747-50.

3. L. Carlitz. "Single Variable Bell Polynomials." CollectaneaMathematica 14 (1962): 13-25. 4. L. Carlitz. "Set Partitions." The Fibonacci Quarterly UA (1916)321-42. 5. L. Carlitz & J. Riordan. "The Divided Central Differences of Zero." Canad. J. Math. 15

(1963):94-100. 6. Ch. A. Charalambides. "Central Factorial Numbers and Related Expansions." The Fibonacci

Quarterly 19,5 (1981):451-56. 7. E. Lehmer. "On Congruences Involving Bernoulli Numbers and the Quotients of Fermat and

Wilson." Annals of Math. 39 (1938):350-60. 8. Barry J. Powell. Advanced Problem 6325. Amer. Math. Monthly 87 (1980):826. 9. J. Riordan. Combinatorial Identities. New York: Wiley, 1968.

AMS Classification Numbers: 11B37, 11B39

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